Group #30639571 — subgroups

Subgroups — #30639571

Order 16256.

396 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549900	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16861126	C₂	1	4	yes	maximal, complemented
3	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	2	yes	—
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
8	#16861126	—	2	8	no	—
9	#16861120	—	2	1	no	—
10	C₄×C₁₂₇⋊C₈	—	2	1	no	—
11	C₂²×D₄×C₁₂₇	C₄	1	4	yes	—
12	C₂²×D₄×C₁₂₇	C₂²	1	2	yes	—
13	#127×C₁₂₇	—	2	1	no	—
14	#127×C₁₂₇	—	2	1	no	—
15	#127×C₁₂₇	—	2	2	no	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	#14523232	—	2	4	no	—
18	#14523232	—	2	2	no	—
19	#14523232	D₄	1	4	yes	complemented
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂⁴×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₁₂₇	—	2	4	no	—
24	C₂×D₄×C₁₂₇	—	2	4	no	—
25	C₂×D₄×C₁₂₇	—	2	2	no	—
26	C₂×D₄×C₁₂₇	—	2	2	no	—
27	C₂×D₄×C₁₂₇	—	2	4	no	—
28	C₂×D₄×C₁₂₇	—	2	4	no	—
29	C₂×D₄×C₁₂₇	—	4	8	no	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
33	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
34	C₄²×C₁₂₇	D₄	1	2	yes	—
35	C₁₂₇⋊C₈	—	2	4	no	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₂₇	—	2	4	no	—
38	C₂³×C₁₂₇	—	2	8	no	—
39	C₂³×C₁₂₇	—	4	8	no	—
40	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₁₂₇	—	4	8	no	—
42	D₄×C₁₂₇	—	4	8	no	—
43	D₄×C₁₂₇	—	4	4	no	—
44	D₄×C₁₂₇	—	4	2	no	—
45	D₄×C₁₂₇	—	4	2	no	—
46	C₂×C₄×C₁₂₇	—	2	4	no	—
47	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	#41	1	2	yes	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	4	4	no	—
55	C₂²×C₁₂₇	—	4	8	no	—
56	C₂²×C₁₂₇	—	8	8	no	—
57	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	#100	1	4	yes	complemented
61	C₄×C₁₂₇	#121	1	2	yes	—
62	C₄×C₁₂₇	#124	1	1	yes	characteristic
63	C₄×C₁₂₇	#127	1	1	yes	characteristic
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	—	8	4	no	—
66	C₂×C₁₂₇	#390	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#100	1	2	yes	—
68	#1965	—	127	1	no	maximal, Sylow
69	C₁₂₇	#1965	1	1	yes	characteristic, complemented, Sylow
70	C₂×#127	D₁₂₇	1	1	yes	characteristic
71	#423	—	127	1	no	—
72	C₂×#100	—	127	4	no	—
73	C₄×M₁₆	—	127	1	no	—
74	C₂²×D₄	Dic₁₂₇	1	4	yes	—
75	C₂²×D₄	C₂×D₁₂₇	1	2	yes	—
76	C₂×M₁₆	—	127	2	no	—
77	C₂×M₁₆	—	127	2	no	—
78	#127	—	2	1	no	—
79	#127	—	2	1	no	—
80	#127	—	2	2	no	—
81	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄⋊C₈	—	254	2	no	—
83	#100	—	254	8	no	—
84	#97	—	254	1	no	—
85	C₄×C₈	—	254	1	no	—
86	C₂⁴	C₂×Dic₁₂₇	1	4	yes	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	4	8	no	—
94	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
96	M₁₆	—	127	4	no	—
97	M₁₆	—	254	4	no	—
98	M₁₆	—	254	2	no	—
99	C₂×C₈	—	254	2	no	—
100	C₂×C₈	—	254	2	no	—
101	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
102	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
103	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	—	2	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
108	D₄	—	4	8	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	8	no	—
112	D₄	—	4	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
116	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	#14523247	1	2	yes	—
120	C₈	—	508	2	no	—
121	C₈	—	254	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	8	no	—
125	C₂²	—	8	8	no	—
126	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
127	C₂²	D₄×Dic₁₂₇	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#16861126	1	4	yes	complemented
130	C₄	#16861180	1	2	yes	—
131	C₄	#16861196	1	1	yes	characteristic
132	C₄	#16861205	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	8	4	no	—
135	C₂	#22549779	1	1	yes	characteristic
136	C₂	C₂×#16861126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549900

C₂

yes

maximal, characteristic, complemented

C₂×#16861126

C₂

yes

maximal, complemented

C₄×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂×#14523232

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

—

#16861126

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₂²×D₄×C₁₂₇

C₄

yes

—

C₂²×D₄×C₁₂₇

C₂²

yes

—

#127×C₁₂₇

—

#127×C₁₂₇

—

#127×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

—

D₄

yes

complemented

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂⁴×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#100

yes

complemented

C₄×C₁₂₇

#121

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

C₂×#100

yes

—

#1965

—

127

maximal, Sylow

C₁₂₇

#1965

yes

characteristic, complemented, Sylow

C₂×#127

D₁₂₇

yes

characteristic

#423

—

127

—

C₂×#100

—

127

—

C₄×M₁₆

—

127

—

C₂²×D₄

Dic₁₂₇

yes

—

C₂²×D₄

C₂×D₁₂₇

yes

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

#100

—

254

—

#97

—

254

—

C₄×C₈

—

254

—

C₂⁴

C₂×Dic₁₂₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

254

—

100

C₂×C₈

—

254

—

101

C₄²

C₂²⋊D₁₂₇

yes

characteristic

102

C₄²

C₂²⋊D₁₂₇

yes

characteristic

103

C₄²

C₂²⋊D₁₂₇

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×Dic₁₂₇

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

115

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

116

C₂×C₄

D₄×D₁₂₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

C₂×C₄

#14523247

yes

—

120

C₈

—

508

—

121

C₈

—

254

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

127

C₂²

D₄×Dic₁₂₇

yes

—

128

C₄

—

129

C₄

#16861126

yes

complemented

130

C₄

#16861180

yes

—

131

C₄

#16861196

yes

characteristic

132

C₄

#16861205

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#22549779

yes

characteristic

136

C₂

C₂×#16861126

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₂₇
commutator	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting, prime radical	C₂×#127×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#127	D₁₂₇
hypocenter	C₁₂₇	#1965
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₂₇

commutator

C₂³×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting, prime radical

C₂×#127×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #22549900	C₂	semidirect
2. C₂×#16861126	C₂	semidirect
3. C₄×#14523232	C₂	semidirect
19. #14523232	D₄	semidirect
60. C₄×C₁₂₇	#100	semidirect
69. C₁₂₇	#1965	semidirect
129. C₄	#16861126	semidirect

H ≅ G/N

type

1. #22549900

C₂

semidirect

2. C₂×#16861126

C₂

semidirect

3. C₄×#14523232

C₂

semidirect

19. #14523232

D₄

semidirect

60. C₄×C₁₂₇

#100

semidirect

69. C₁₂₇

#1965

semidirect

129. C₄

#16861126

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22549900	2
C₂×#16861126	2
C₄×#14523232	2
C₂×#127×C₁₂₇	2
#1965	127

subgroup

index

C₂×#127×C₁₂₇

127

Subgroups — #30639571

Conjugacy classes of subgroups

Complements

Maximal subgroups